AQA FP3 (Further Pure Mathematics 3) 2008 June

1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln ( x + y )$$ and $$y ( 2 ) = 3$$ Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 2.1 )\), giving your answer to four decimal places.
(6 marks)

2

1. Find the values of the constants \(a , b , c\) and \(d\) for which \(a + b x + c \sin x + d \cos x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 10 \sin x - 3 x$$ (4 marks)
2. Hence find the general solution of this differential equation.

3

1. Show that \(x ^ { 2 } = 1 - 2 y\) can be written in the form \(x ^ { 2 } + y ^ { 2 } = ( 1 - y ) ^ { 2 }\).
2. A curve has cartesian equation \(x ^ { 2 } = 1 - 2 y\). Find its polar equation in the form \(r = \mathrm { f } ( \theta )\), given that \(r > 0\).

4

1. A differential equation is given by $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
2. By using an integrating factor, find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
3. Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$ giving your answer in the form \(y = \mathrm { g } ( x )\).

5

1. Find \(\int x ^ { 3 } \ln x \mathrm {~d} x\).
2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\) is an improper integral.
3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\), showing the limiting process used.

6

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 10 \mathrm { e } ^ { - 2 x } - 9$$ (10 marks)
2. Hence express \(y\) in terms of \(x\), given that \(y = 7\) when \(x = 0\) and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow 0\) as \(x \rightarrow \infty\).

7

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
2. 1. Given that \(y = \sqrt { 3 + \mathrm { e } ^ { x } }\), find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\).
   2. Using Maclaurin's theorem, show that, for small values of \(x\), $$\sqrt { 3 + \mathrm { e } ^ { x } } \approx 2 + \frac { 1 } { 4 } x + \frac { 7 } { 64 } x ^ { 2 }$$
3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 3 + \mathrm { e } ^ { x } } - 2 } { \sin 2 x } \right]$$

8 The polar equation of a curve \(C\) is $$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$

1. Verify that the points \(A\) and \(B\), with polar coordinates ( 7,0 ) and ( \(3 , \pi\) ) respectively, lie on the curve \(C\).
2. Sketch the curve \(C\).
3. Find the area of the region bounded by the curve \(C\).
4. The point \(P\) is the point on the curve \(C\) for which \(\theta = \alpha\), where \(0 < \alpha \leqslant \frac { \pi } { 2 }\). The point \(Q\) lies on the curve such that \(P O Q\) is a straight line, where the point \(O\) is the pole. Find, in terms of \(\alpha\), the area of triangle \(O Q B\).